Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 9"
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If <math>x=\sqrt[3]{4}</math> and <math>y=\sqrt[3]{6}-\sqrt[3]{3}</math>, then which of the following is correct? | If <math>x=\sqrt[3]{4}</math> and <math>y=\sqrt[3]{6}-\sqrt[3]{3}</math>, then which of the following is correct? | ||
− | + | <math>\mathrm{(A)}\ x=y\qquad\mathrm{(B)}\ x<y\qquad\mathrm{(C)}\ x=2y\qquad\mathrm{(D)}\ x>2y\qquad\mathrm{(E)}\ \text{None of these}</math> | |
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− | B | ||
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− | C | ||
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− | D | ||
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==Solution== | ==Solution== | ||
− | The question is asking us for an approximation of the [[ratio]] between <math>x : y</math>. Thus we are allowed to multiply both sides by a | + | The question is asking us for an approximation of the [[ratio]] between <math>x : y</math>. Thus, we are allowed to [[multiply]] both sides by a [[constant]]. |
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+ | By [[difference of cube]]s, | ||
+ | <cmath> | ||
+ | \begin{align*}\sqrt[3]{4}(\sqrt[3]{36}+\sqrt[3]{18}+\sqrt[3]{9})&:(\sqrt[3]{6}-\sqrt[3]{3})(\sqrt[3]{36}+\sqrt[3]{18}+\sqrt[3]{9})\\ | ||
+ | 2\sqrt[3]{18}+2\sqrt[3]{9}+\sqrt[3]{36}&:3\end{align*} | ||
+ | </cmath> | ||
We can approximate the terms on the LHS; <math>2\sqrt[3]{18} > 4</math>, <math>2\sqrt[3]{9} > 4</math>, <math>\sqrt[3]{36} > 3</math>, so the sum on the left side <math>> 11</math>. Hence <math>x > 2y</math>, and the answer is <math>\mathrm{(D)}</math>. | We can approximate the terms on the LHS; <math>2\sqrt[3]{18} > 4</math>, <math>2\sqrt[3]{9} > 4</math>, <math>\sqrt[3]{36} > 3</math>, so the sum on the left side <math>> 11</math>. Hence <math>x > 2y</math>, and the answer is <math>\mathrm{(D)}</math>. | ||
Latest revision as of 10:40, 27 April 2008
Problem
If and
, then which of the following is correct?
Solution
The question is asking us for an approximation of the ratio between . Thus, we are allowed to multiply both sides by a constant.
By difference of cubes,
We can approximate the terms on the LHS;
,
,
, so the sum on the left side
. Hence
, and the answer is
.
Remark: There doesn't seem to be any direct way to calculate a simple ratio between the two terms, but various variations can involve approximating terms by multiplying by certain quantities.
See also
2006 Cyprus MO, Lyceum (Problems) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 |